Conclusión: Se concluye que los coeficientes \(\beta_0\), \(\beta_1\), \(\beta_2\), \(\beta_3\), \(\beta_4\), \(\beta_8\), \(\beta_12\) correspondientes a los tratamientos B, C, D y E, a la fila 5 y columna 5 respectivamente, fueron estadísticamente distintos de 0. El ajuste del modelo descrito en el \(R^2\) fue de 0.921, lo que quiere decir que el 92.1 % de la varianza del rendimiento está explicado por las variables Tratamiento, Fila y Columna.
Durbin-Watson test
data: modelo.dcl
DW = 1.753, p-value = 0.2041
alternative hypothesis: true autocorrelation is not 0
Conclusión. A un nivel de significancia de 0.1, se concluye que no existe suficiente evidencia estadística para rechazar la hipótesis nula, por lo tanto, los residuos del rendimiento son completamente aleatorios e independientes.
Normalidad de residuos
\(H_0: \text{La distribución de los residuos del rendimiento es similar a la función normal}\)
\(H_1: \text{La distribución de los residuos del rendimiento es similar a la función normal}\)
shapiro.test(rstudent(modelo.dcl))
Shapiro-Wilk normality test
data: rstudent(modelo.dcl)
W = 0.96303, p-value = 0.4782
nortest::ad.test(rstudent(modelo.dcl))
Anderson-Darling normality test
data: rstudent(modelo.dcl)
A = 0.38392, p-value = 0.3688
lillie.test(rstudent(modelo.dcl))
Lilliefors (Kolmogorov-Smirnov) normality test
data: rstudent(modelo.dcl)
D = 0.13219, p-value = 0.3133
Exact one-sample Kolmogorov-Smirnov test
data: rstudent(modelo.dcl)
D = 0.10352, p-value = 0.9264
alternative hypothesis: two-sided
cvm.test(rstudent(modelo.dcl))
Cramer-von Mises normality test
data: rstudent(modelo.dcl)
W = 0.054267, p-value = 0.4357
pearson.test(rstudent(modelo.dcl))
Pearson chi-square normality test
data: rstudent(modelo.dcl)
P = 7.96, p-value = 0.1585
sf.test(rstudent(modelo.dcl))
Shapiro-Francia normality test
data: rstudent(modelo.dcl)
W = 0.95883, p-value = 0.3324
Conclusión. A un nivel de significancia de 0.1, se concluye que no existe suficiente evidencia estadística para rechazar la hipótesis nula, por lo tanto, la distribución de los residuos del rendimiento es similar a la función normal o gaussiana.
Homocedasticidad
\(H_0\): La varianza del rendimiento es constante con respecto a los valores ajustados del rendimiento.
\(H_1\): La varianza del rendimiento no es constante con respecto a los valores ajustados del rendimiento.
ncvTest(modelo.dcl)
Non-constant Variance Score Test
Variance formula: ~ fitted.values
Chisquare = 0.915634, Df = 1, p = 0.33862
bptest(modelo.dcl)
studentized Breusch-Pagan test
data: modelo.dcl
BP = 15.008, df = 12, p-value = 0.241
bptest(modelo.dcl, studentize = F)
Breusch-Pagan test
data: modelo.dcl
BP = 12.906, df = 12, p-value = 0.3759
olsrr::ols_test_breusch_pagan(modelo.dcl)
Breusch Pagan Test for Heteroskedasticity
-----------------------------------------
Ho: the variance is constant
Ha: the variance is not constant
Data
-------------------------------
Response : rdt
Variables: fitted values of rdt
Test Summary
---------------------------
DF = 1
Chi2 = 0.915634
Prob > Chi2 = 0.338624
Conclusión. A un nivel de significancia de 0.1, se concluye que no existe suficiente evidencia estadística para rechazar la hipótesis nula, por lo tanto, la varianza del rendimiento es constante con respecto a los valores ajustados del rendimiento.
Recomendación. Debido a que se cumple con el supuesto de homocedasticidad, para evaluar los efectos de los tratamientos con respecto al rendimiento, se debe proceder a realizar el análisis de varianza.
Estadísticas globales
modelo.dcl %>%gvlma()
Call:
lm(formula = rdt ~ trt + F + C, data = data)
Coefficients:
(Intercept) trtB trtC trtD trtE FF2
44.670 4.584 8.594 12.034 11.744 2.152
FF3 FF4 FF5 CC2 CC3 CC4
-1.864 -1.576 3.154 1.862 2.006 2.864
CC5
3.724
ASSESSMENT OF THE LINEAR MODEL ASSUMPTIONS
USING THE GLOBAL TEST ON 4 DEGREES-OF-FREEDOM:
Level of Significance = 0.05
Call:
gvlma(x = .)
Value p-value Decision
Global Stat 3.3051 0.5081 Assumptions acceptable.
Skewness 0.3324 0.5642 Assumptions acceptable.
Kurtosis 0.0817 0.7750 Assumptions acceptable.
Link Function 1.7360 0.1876 Assumptions acceptable.
Heteroscedasticity 1.1550 0.2825 Assumptions acceptable.
Conclusión. A un nivel de significancia de 0.05, se concluye que existe suficiente evidencia estadística para rechazar la hipótesis nula, por lo tanto, al menos un tratamiento tiene un efecto sobre el rendimiento estadísticamente diferente del resto de tratamientos.
agricolae::cv.model(modelo.dcl)
[1] 3.984179
Comparaciones de medias
A vs B:
\(H_0: \mu_{A} - \mu_{B} = 0\)
\(H_1: \mu_{A} - \mu_{B} \neq 0\)
A vs C:
\(H_0: \mu_{A} - \mu_{C} = 0\)
\(H_1: \mu_{A} - \mu_{C} \neq 0\)
A vs D:
\(H_0: \mu_{A} - \mu_{D} = 0\)
\(H_1: \mu_{A} - \mu_{D} \neq 0\)
A vs E:
\(H_0: \mu_{A} - \mu_{E} = 0\)
\(H_1: \mu_{A} - \mu_{E} \neq 0\)
B vs C:
\(H_0: \mu_{B} - \mu_{C} = 0\)
\(H_1: \mu_{B} - \mu_{C} \neq 0\)
B vs D:
\(H_0: \mu_{B} - \mu_{D} = 0\)
\(H_1: \mu_{B} - \mu_{D} \neq 0\)
B vs E:
\(H_0: \mu_{B} - \mu_{E} = 0\)
\(H_1: \mu_{B} - \mu_{E} \neq 0\)
C vs D:
\(H_0: \mu_{C} - \mu_{D} = 0\)
\(H_1: \mu_{C} - \mu_{D} \neq 0\)
C vs E:
\(H_0: \mu_{C} - \mu_{E} = 0\)
\(H_1: \mu_{C} - \mu_{E} \neq 0\)
D vs E:
\(H_0: \mu_{D} - \mu_{E} = 0\)
\(H_1: \mu_{D} - \mu_{E} \neq 0\)
Prueba de HSD (Honestamente significativa -> Tukey)
agricolae::HSD.test(modelo.dcl, trt ="trt", alpha =0.05,group =TRUE, main =NULL, console =TRUE)
Study: modelo.dcl ~ "trt"
HSD Test for rdt
Mean Square Error: 4.719243
trt, means
rdt std r Min Max
A 47.134 2.840129 5 42.26 49.43
B 51.718 5.324328 5 44.41 57.31
C 55.728 2.229747 5 53.72 59.45
D 59.168 2.065798 5 55.87 60.89
E 58.878 1.710181 5 55.87 60.17
Alpha: 0.05 ; DF Error: 12
Critical Value of Studentized Range: 4.50771
Minimun Significant Difference: 4.379324
Treatments with the same letter are not significantly different.
rdt groups
D 59.168 a
E 58.878 a
C 55.728 ab
B 51.718 b
A 47.134 c
agricolae::HSD.test(modelo.dcl, trt ="trt", alpha =0.05,group =FALSE, main =NULL, console =TRUE)
Study: modelo.dcl ~ "trt"
HSD Test for rdt
Mean Square Error: 4.719243
trt, means
rdt std r Min Max
A 47.134 2.840129 5 42.26 49.43
B 51.718 5.324328 5 44.41 57.31
C 55.728 2.229747 5 53.72 59.45
D 59.168 2.065798 5 55.87 60.89
E 58.878 1.710181 5 55.87 60.17
Alpha: 0.05 ; DF Error: 12
Critical Value of Studentized Range: 4.50771
Comparison between treatments means
difference pvalue signif. LCL UCL
A - B -4.584 0.0388 * -8.963324 -0.2046756
A - C -8.594 0.0003 *** -12.973324 -4.2146756
A - D -12.034 0.0000 *** -16.413324 -7.6546756
A - E -11.744 0.0000 *** -16.123324 -7.3646756
B - C -4.010 0.0786 . -8.389324 0.3693244
B - D -7.450 0.0012 ** -11.829324 -3.0706756
B - E -7.160 0.0017 ** -11.539324 -2.7806756
C - D -3.440 0.1539 -7.819324 0.9393244
C - E -3.150 0.2126 -7.529324 1.2293244
D - E 0.290 0.9995 -4.089324 4.6693244
Nota: Todos aquellos tratamientos que compartan por lo menos una letra serán estadísticamente similares.
Recomendación: Se debe usar la prueba de Tukey cuando como máximo se tenga 6 tratamientos.
Prueba de Duncan
agricolae::duncan.test(modelo.dcl, trt ="trt", alpha =0.05, group =TRUE, main =NULL, console =TRUE)
Study: modelo.dcl ~ "trt"
Duncan's new multiple range test
for rdt
Mean Square Error: 4.719243
trt, means
rdt std r Min Max
A 47.134 2.840129 5 42.26 49.43
B 51.718 5.324328 5 44.41 57.31
C 55.728 2.229747 5 53.72 59.45
D 59.168 2.065798 5 55.87 60.89
E 58.878 1.710181 5 55.87 60.17
Alpha: 0.05 ; DF Error: 12
Critical Range
2 3 4 5
2.993547 3.133384 3.218110 3.274185
Means with the same letter are not significantly different.
rdt groups
D 59.168 a
E 58.878 a
C 55.728 b
B 51.718 c
A 47.134 d
agricolae::duncan.test(modelo.dcl, trt ="trt", alpha =0.05, group =FALSE, main =NULL, console =TRUE)
Study: modelo.dcl ~ "trt"
Duncan's new multiple range test
for rdt
Mean Square Error: 4.719243
trt, means
rdt std r Min Max
A 47.134 2.840129 5 42.26 49.43
B 51.718 5.324328 5 44.41 57.31
C 55.728 2.229747 5 53.72 59.45
D 59.168 2.065798 5 55.87 60.89
E 58.878 1.710181 5 55.87 60.17
Comparison between treatments means
difference pvalue signif. LCL UCL
A - B -4.584 0.0059 ** -7.577547 -1.5904531
A - C -8.594 0.0001 *** -11.727384 -5.4606157
A - D -12.034 0.0000 *** -15.308185 -8.7598153
A - E -11.744 0.0000 *** -14.962110 -8.5258900
B - C -4.010 0.0129 * -7.003547 -1.0164531
B - D -7.450 0.0003 *** -10.668110 -4.2318900
B - E -7.160 0.0003 *** -10.293384 -4.0266157
C - D -3.440 0.0339 * -6.573384 -0.3066157
C - E -3.150 0.0407 * -6.143547 -0.1564531
D - E 0.290 0.8364 -2.703547 3.2835469
Nota: La prueba de Duncan usa múltiples valores críticos.
Recomendación: No usar la prueba de Duncan cuando se presentan resultados con múltiples tratamientos que obtengan varias significancias.
Prueba de Student - Newman -Keuls
agricolae::SNK.test(modelo.dcl, trt ="trt", alpha =0.05, group =TRUE, main =NULL, console =TRUE)
Study: modelo.dcl ~ "trt"
Student Newman Keuls Test
for rdt
Mean Square Error: 4.719243
trt, means
rdt std r Min Max
A 47.134 2.840129 5 42.26 49.43
B 51.718 5.324328 5 44.41 57.31
C 55.728 2.229747 5 53.72 59.45
D 59.168 2.065798 5 55.87 60.89
E 58.878 1.710181 5 55.87 60.17
Alpha: 0.05 ; DF Error: 12
Critical Range
2 3 4 5
2.993547 3.665471 4.079077 4.379324
Means with the same letter are not significantly different.
rdt groups
D 59.168 a
E 58.878 a
C 55.728 a
B 51.718 b
A 47.134 c
agricolae::SNK.test(modelo.dcl, trt ="trt", alpha =0.05, group =FALSE, main =NULL, console =TRUE)
Study: modelo.dcl ~ "trt"
Student Newman Keuls Test
for rdt
Mean Square Error: 4.719243
trt, means
rdt std r Min Max
A 47.134 2.840129 5 42.26 49.43
B 51.718 5.324328 5 44.41 57.31
C 55.728 2.229747 5 53.72 59.45
D 59.168 2.065798 5 55.87 60.89
E 58.878 1.710181 5 55.87 60.17
Comparison between treatments means
difference pvalue signif. LCL UCL
A - B -4.584 0.0059 ** -7.577547 -1.5904531
A - C -8.594 0.0001 *** -12.259471 -4.9285291
A - D -12.034 0.0000 *** -16.413324 -7.6546756
A - E -11.744 0.0000 *** -15.823077 -7.6649232
B - C -4.010 0.0129 * -7.003547 -1.0164531
B - D -7.450 0.0008 *** -11.529077 -3.3709232
B - E -7.160 0.0006 *** -10.825471 -3.4945291
C - D -3.440 0.0666 . -7.105471 0.2254709
C - E -3.150 0.0407 * -6.143547 -0.1564531
D - E 0.290 0.8364 -2.703547 3.2835469
Prueba de Least significant difference
agricolae::LSD.test(modelo.dcl, trt ="trt", alpha =0.05, group =TRUE, main =NULL, console =TRUE)
Study: modelo.dcl ~ "trt"
LSD t Test for rdt
Mean Square Error: 4.719243
trt, means and individual ( 95 %) CI
rdt std r LCL UCL Min Max
A 47.134 2.840129 5 45.01724 49.25076 42.26 49.43
B 51.718 5.324328 5 49.60124 53.83476 44.41 57.31
C 55.728 2.229747 5 53.61124 57.84476 53.72 59.45
D 59.168 2.065798 5 57.05124 61.28476 55.87 60.89
E 58.878 1.710181 5 56.76124 60.99476 55.87 60.17
Alpha: 0.05 ; DF Error: 12
Critical Value of t: 2.178813
least Significant Difference: 2.993547
Treatments with the same letter are not significantly different.
rdt groups
D 59.168 a
E 58.878 a
C 55.728 b
B 51.718 c
A 47.134 d
agricolae::LSD.test(modelo.dcl, trt ="trt", alpha =0.05, group =FALSE, main =NULL, console =TRUE)
Study: modelo.dcl ~ "trt"
LSD t Test for rdt
Mean Square Error: 4.719243
trt, means and individual ( 95 %) CI
rdt std r LCL UCL Min Max
A 47.134 2.840129 5 45.01724 49.25076 42.26 49.43
B 51.718 5.324328 5 49.60124 53.83476 44.41 57.31
C 55.728 2.229747 5 53.61124 57.84476 53.72 59.45
D 59.168 2.065798 5 57.05124 61.28476 55.87 60.89
E 58.878 1.710181 5 56.76124 60.99476 55.87 60.17
Alpha: 0.05 ; DF Error: 12
Critical Value of t: 2.178813
Comparison between treatments means
difference pvalue signif. LCL UCL
A - B -4.584 0.0059 ** -7.577547 -1.590453
A - C -8.594 0.0000 *** -11.587547 -5.600453
A - D -12.034 0.0000 *** -15.027547 -9.040453
A - E -11.744 0.0000 *** -14.737547 -8.750453
B - C -4.010 0.0129 * -7.003547 -1.016453
B - D -7.450 0.0002 *** -10.443547 -4.456453
B - E -7.160 0.0002 *** -10.153547 -4.166453
C - D -3.440 0.0277 * -6.433547 -0.446453
C - E -3.150 0.0407 * -6.143547 -0.156453
D - E 0.290 0.8364 -2.703547 3.283547
------------------------------------------------------------------------
Analysis of Variance Table
------------------------------------------------------------------------
DF SS MS Fc Pr>Fc
Treatament 4 522.30 130.574 13.439 1.7811e-05
Residuals 20 194.32 9.716
Total 24 716.61
------------------------------------------------------------------------
CV = 5.72 %
------------------------------------------------------------------------
Shapiro-Wilk normality test
p-value: 0.6338902
According to Shapiro-Wilk normality test at 5% of significance, residuals can be considered normal.
------------------------------------------------------------------------
------------------------------------------------------------------------
Homogeneity of variances test
p-value: 0.1038307
According to the test of levene at 5% of significance, residuals can be considered homocedastic.
------------------------------------------------------------------------
Test of Calinski & Corsten based on F distribution
------------------------------------------------------------------------
Groups Treatments Means
4 a D 59.168
5 a E 58.878
3 a C 55.728
2 b B 51.718
1 b A 47.134
------------------------------------------------------------------------
------------------------------------------------------------------------
Analysis of Variance Table
------------------------------------------------------------------------
DF SS MS Fc Pr>Fc
Treatament 4 522.30 130.574 21.9656 0.0000025
Block 4 99.20 24.801 4.1721 0.0167511
Residuals 16 95.11 5.944
Total 24 716.61
------------------------------------------------------------------------
CV = 4.47 %
------------------------------------------------------------------------
Shapiro-Wilk normality test
p-value: 0.4781707
According to Shapiro-Wilk normality test at 5% of significance, residuals can be considered normal.
------------------------------------------------------------------------
------------------------------------------------------------------------
Homogeneity of variances test
p-value: 0.02089805
WARNING: at 5% of significance, residuals can not be considered homocedastic!
------------------------------------------------------------------------
Tukey's test
------------------------------------------------------------------------
Groups Treatments Means
a D 59.168
a E 58.878
ab C 55.728
bc B 51.718
c A 47.134
------------------------------------------------------------------------
------------------------------------------------------------------------
Analysis of Variance Table
------------------------------------------------------------------------
DF SS MS Fc Pr>Fc
Treatament 4 522.30 130.574 27.6685 0.000006
Row 4 99.20 24.801 5.2553 0.011101
Column 4 38.48 9.620 2.0385 0.152720
Residuals 12 56.63 4.719
Total 24 716.61
------------------------------------------------------------------------
CV = 3.98 %
------------------------------------------------------------------------
Shapiro-Wilk normality test
p-value: 0.6922026
According to Shapiro-Wilk normality test at 5% of significance, residuals can be considered normal.
------------------------------------------------------------------------
Scott-Knott test
------------------------------------------------------------------------
Groups Treatments Means
1 a D 59.168
2 a E 58.878
3 b C 55.728
4 c B 51.718
5 d A 47.134
------------------------------------------------------------------------
Se observa que cada método posee una distribución distinta del resto de distribuciones de los métodos.
\(H_0: \text{Las observaciones de los métodos provienen de la misma distribución}\)
\(H_1: \text{Las observaciones de los métodos no provienen de la misma distribución}\)
corn %>%ad.test(observation~method)
Anderson-Darling k-sample test.
Number of samples: 3
Sample sizes: 34, 34, 34
Number of ties: 64
Mean of Anderson-Darling Criterion: 2
Standard deviation of Anderson-Darling Criterion: 1.05185
T.AD = ( Anderson-Darling Criterion - mean)/sigma
Null Hypothesis: All samples come from a common population.
AD T.AD asympt. P-value
version 1: 54.23 49.66 1.150e-30
version 2: 54.60 49.98 5.614e-31
Conclusión: A un nivel de significancia de 0.05, se concluye que existe suficiente evidencia estadística para rechazar la hipótesis nula. Por lo tanto, las observaciones de los métodos no provienen de la misma distribución.
Recomendación: En base a la evidencia, no se recomienda usar la prueba de Kruskall Wallis.
The Median Test for observation ~ method
Chi Square = 17.54306 DF = 3 P.Value 0.00054637
Median = 89
Median r Min Max Q25 Q75
1 91.0 9 83 96 89.00 92.00
2 86.0 10 81 91 83.25 89.75
3 95.0 7 91 101 93.50 98.00
4 80.5 8 77 82 78.75 81.00
Post Hoc Analysis
Groups according to probability of treatment differences and alpha level.
Treatments with the same letter are not significantly different.
observation groups
3 95.0 a
1 91.0 b
2 86.0 b
4 80.5 c
out$statistics
Chisq Df p.chisq Median
17.54306 3 0.00054637 89
out$parameters
test name.t ntr alpha
Median method 4 0.05
out$medians
Median r Min Max Q25 Q75
1 91.0 9 83 96 89.00 92.00
2 86.0 10 81 91 83.25 89.75
3 95.0 7 91 101 93.50 98.00
4 80.5 8 77 82 78.75 81.00
out$groups
observation groups
3 95.0 a
1 91.0 b
2 86.0 b
4 80.5 c
Y <- grass$evaluationtratamiento <- grass$trtbloque <- grass$judgeAnp <-friedman(judge = bloque,trt = tratamiento, evaluation = Y, alpha=0.05, group=TRUE, console=TRUE, main=NULL)
Study: Y ~ bloque + tratamiento
tratamiento, Sum of the ranks
Y r
t1 38.0 12
t2 23.5 12
t3 24.5 12
t4 34.0 12
Friedman's Test
===============
Adjusted for ties
Critical Value: 8.097345
P.Value Chisq: 0.04404214
F Value: 3.192198
P.Value F: 0.03621547
Post Hoc Analysis
Alpha: 0.05 ; DF Error: 33
t-Student: 2.034515
LSD: 11.48168
Treatments with the same letter are not significantly different.
Sum of ranks groups
t1 38.0 a
t4 34.0 ab
t3 24.5 b
t2 23.5 b